If $A=(x: x=4 n, n \in Z)$ and $B=\{x: x=6 n, n \in Z\}$, then $A \cap B=$

$\left\{ {x:x = 10n,n \in Z} \right\}$

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JEE • MATHEMATICS • SET THEORY

Set theory practice · Easy

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Showing 1-30 of 194 questions

1 Easy

1 of 194

Understanding +4 | -1

Which of the following cannot be the number of elements in the power set of any finite set?

View answer and explanation

Correct option: A

Option A (26) is correct because the number of elements in a power set must be a power of 2, and 26 is not a power of 2.

2 Easy

2 of 194

Application +4 | -1

In a town of 10000 families, it was found that

40 \%

families buy newspaper

A, 20 \%

families buy newspaper B and

10 \%

families buy newspaper

C

, 5% families buy newspapers

A

and B, 3% buy newspapers B and

C

and

4 \%

buy newspapers

A

and

C

. If

2 \%

families buy all the three newspapers, then number of families which buy

A

only is

View answer and explanation

Correct option: B

3300

Option B is correct because by using the principle of inclusion-exclusion, the number of families buying newspaper A only is calculated to be 3300.

3 Easy

3 of 194

Application +4 | -1

A

and

B

are two sets such that

A

has 12 elements,

B

has 17 elements and

A \cup B

has 21 elements, then number of elements in

{A} \cap {B}

are

View answer and explanation

Correct option: C

Option C (8) is correct because it correctly uses the inclusion-exclusion principle to find the number of elements in the intersection of sets A and B.

4 Easy

4 of 194

Analyzing +4 | -1

Let

A = \left\{ {x \in R:|x + 1| < 2} \right\}{\text{ }}\,\,{\text{and }}\,\,\,B = \left\{ {x \in R:|x - 1| \geqslant 2} \right\}.

Then which one of the following statements is NOT true?

View answer and explanation

Correct option: B

B - A = R - ( - 3,1)

The statement

B - A = R - ( - 3,1)

is NOT true because

B - A

should be

R - (-3, 3)

. Therefore, option B is the correct answer.

5 Easy

5 of 194

Understanding +4 | -1

If the set

{A}

has 3 elements and the set

{B}=\{3,4\}

, then the number of elements in

{A} \times {B}

View answer and explanation

Correct option: A

6 is the correct answer because the number of elements in the Cartesian product A x B is the product of the number of elements in A and B, which is 3 * 2 = 6.

6 Easy

6 of 194

Application +4 | -1

{X}

and

{Y}

are two sets such that

{X} \cup {Y}

has 18 elements,

{X}

has 8 elements and

{Y}

has 15 elements; then number of elements in

{X} \cap {Y}

are

View answer and explanation

Correct option: A

Option A (5) is correct because it accurately applies the inclusion-exclusion principle to determine the number of elements in the intersection of sets X and Y.

7 Easy

7 of 194

Application +4 | -1

If the sets

A

and

B

are defined as

A=\{(x, y): y

\left.=\frac{1}{x}, 0 \neq x \in R\right\} B=\{(x, y): y=-x, x \in R\}

, then:

View answer and explanation

Correct option: C

A \cap B=\phi

Option C (

A \cap B=\phi

) is correct because the equations

y=\frac{1}{x}

and

y=-x

have no real solutions, meaning the sets have no common elements.

8 Easy

8 of 194

Understanding +4 | -1

A=(x: x=4 n, n \in Z)

and

B=\{x: x=6 n, n \in Z\}

, then

A \cap B=

View answer and explanation

Correct option: C

\{x: x=12 n, n \in Z\}

Option C (

\{x: x=12 n, n \in Z\}

) is correct because the intersection of the two sets contains elements that are multiples of both 4 and 6, thus multiples of 12.

9 Easy

9 of 194

Understanding +4 | -1

A=\{x: x=2 n+1, n \in Z

and

B=\{x: x=2 n, n \in Z\}

, then

A \cup B

View answer and explanation

Correct option: C

set of integers

Option C (set of integers) is correct because the union of the set of odd integers and the set of even integers is the set of all integers.

10 Easy

10 of 194

Understanding +4 | -1

B

is the set whose elements are obtained by adding 1 to each of the even numbers, then the set builder notation of

B

View answer and explanation

Correct option: C

B=\{x: x

is odd and

x \in Z\}

Option C (

B=\{x: x

is odd and

x \in Z\}

) is correct because adding 1 to each even number results in odd numbers, and the set B contains all odd integers.

11 Easy

11 of 194

Understanding +4 | -1

Which of the following relations is/are correct?

(a)

A-B=A-(A \cap B)

(b)

A=(A \cap B) \cup(A-B)

(c)

A-(B \cup C)=(A-B) \cup(A-C)

View answer and explanation

Correct option: D

(a) and (b)

Option D ((a) and (b)) is correct because the set identities A - B = A - (A ∩ B) and A = (A ∩ B) ∪ (A - B) are both correct.

12 Easy

12 of 194

Understanding +4 | -1

{A}

has 3 elements and

{B}

has 6 elements, then the minimum number of elements in the set

A \cup B

View answer and explanation

Correct option: A

Option A (6) is correct because the minimum number of elements in the union of two sets occurs when one set is a subset of the other.

13 Easy

13 of 194

Application +4 | -1

Suppose

A_1, A_2, \ldots A_{30}

are thirty sets, each with five elements and

{B}_1, {~B}_2, \ldots, {B}_n

are

n

sets each with three elements Let

\bigcup_{i=1}^{30} {~A}_i=\bigcup_{j=1}^n {~B}_j={S}

If each elements of

{S}

belongs to exactly ten of the

{A}_i

's and exactly nine of the

{B}_j

's then

n=

View answer and explanation

Correct option: A

Option A (45) is correct because it equates the total number of elements counted in two different ways using the given information about the sets.

14 Easy

14 of 194

Understanding +4 | -1

A, B, C

are any three sets, then

A \times(B \cup C)

is equal to

View answer and explanation

Correct option: A

(A \times B) \cup(A \times C)

Option A (

(A \times B) \cup (A \times C)

) is correct based on the distributive property of the Cartesian product over union.

15 Easy

15 of 194

Application +4 | -1

Let

A = \left\{ {1,\;2,\;3,\;4} \right\},\;B = \left\{ {2,\;4,\;6} \right\}

Then, the number of sets

C

such that

A \cap B \subseteq C \subseteq A\; \cup B

View answer and explanation

Correct option: C

Option C (8) is correct because the number of sets C is determined by the number of possible combinations of elements from A ∪ B that include A ∩ B.

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16 Easy

16 of 194

Understanding +4 | -1

{P}, {Q}

and

{R}

are subsets of a set

{A}

, then

{R} \times\left({P}^c \cup {Q}^c\right)^c

equals

View answer and explanation

Correct option: A

({R} \times {P}) \cap({R} \times {Q})

Option A ((

R \times P) \cap(R \times Q)

) is correct because it follows from De Morgan's law and the properties of Cartesian products.

17 Easy

17 of 194

Understanding +4 | -1

A=\{a, b\}, B=\{c, d\}, C=\{d, e\},

then

\{(a, c),(a, d),(a, e),(b, c),(b, d),(b, e)\}

is equal to

View answer and explanation

Correct option: C

A \times(B \cup C)

Option C (

A \times (B \cup C)

) is correct because it represents the Cartesian product of A with the union of B and C, resulting in the given set of ordered pairs.

18 Easy

18 of 194

Application +4 | -1

In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number newspapers is

View answer and explanation

Correct option: C

exactly 25

Option C (exactly 25) is correct because the total number of readings is the number of students times the number of newspapers each student reads, which equals the number of newspapers times the number of students reading each newspaper.

19 Easy

19 of 194

Understanding +4 | -1

A=\{1,2,5\}

and

B=\{3,4,5,9\}

, then number of elements in

A \Delta B

is equal to

View answer and explanation

Correct option: A

Option A (5) is correct because A Δ B contains elements in A or B but not in both, resulting in {1, 2, 3, 4, 9}, which has 5 elements.

20 Easy

20 of 194

Understanding +4 | -1

R

and

R^{\prime}

are symmetric relations (not disjoint) on a set

A

, then the relation

R \cap R^{\prime}

View answer and explanation

Correct option: B

Symmetric

Option B (Symmetric) is correct because the intersection of two symmetric relations is also symmetric.

21 Easy

21 of 194

Analyzing +4 | -1

A,B

and

C

are three non-empty sets such that

A

and

B

are disjoint and the number of elements contained in

A

is equal to those contained in the set of elements common to the sets

A

and

C

, then

n\left( {A \cup B \cup C} \right)

is necessarily equal to

View answer and explanation

Correct option: A

n\left( {B \cup C} \right)

Option A is correct because, given the conditions, the number of elements in the union of A, B, and C is equal to the number of elements in the union of B and C.

22 Easy

22 of 194

Application +4 | -1

n\left( u \right) = 100,n\left( A \right) = 50,\;n\left( B \right) = 20

and

n\left( {A \cap B} \right) = 10,

then

n\left\{ {{{\left( {A \cup B} \right)}^c}} \right\}

View answer and explanation

Correct option: C

The correct answer is C (40) because it is calculated by subtracting the number of elements in the union of A and B from the total number of elements in the universal set.

23 Easy

23 of 194

Remembering +4 | -1

If a set

A=\{a, b, c\}

then the number of subsets of the set

A

View answer and explanation

Correct option: C

Option C (8) is correct because a set with n elements has

2^n

subsets, and in this case, n=3, so

2^3 = 8

24 Easy

24 of 194

Analyzing +4 | -1

Let

A

and

B

be two sets that

A \cap X=B \cap X=\phi

and

A \cup X=B \cup X

for some set

X

. Then,

View answer and explanation

Correct option: A

A=B

Option A (

A=B

) is correct because with the given conditions

A \cap X = B \cap X = \phi

and

A \cup X = B \cup X

, it can be shown that set A is equal to set B.

25 Easy

25 of 194

Understanding +4 | -1

A

and

B

are two sets, then

A \cap \left( {A \cup B} \right)

equals

View answer and explanation

Correct option: A

A

Option A (

A

) is correct because

A \cap (A \cup B)

simplifies to A, as A is a subset of

A \cup B

26 Easy

26 of 194

Analyzing +4 | -1

In a certain town 25% families own a phone and 15% own a car, 65% families own neither a phone nor a car. 2000 families own both a car and a phone.

Consider the following statements in this regard:

1. 10 % families own both a car and a phone

2. 35 % families own either a car or a phone

3. 40,000 families live in the town

Which of the following statements are correct?

View answer and explanation

Correct option: C

2 and 3

Option C (2 and 3) is correct because after calculating the percentages and total families, statements 2 and 3 are true, while statement 1 is false.

27 Easy

27 of 194

Understanding +4 | -1

Let

A

be a set represented by the squares of natural number and

x,y

are any two elements of

A

. Then,

View answer and explanation

Correct option: B

xy \in A

Option B (

xy \in A

) is correct because if x and y are squares of natural numbers, their product is also a square of a natural number.

28 Easy

28 of 194

Understanding +4 | -1

A=\left\{(x, y): x^2+y^2 \leq 1, x, y \in R\right\}

and

B=\left\{(x, y): x^2+y^2 \leq 4 ; x, y \in R\right\}

, then

View answer and explanation

Correct option: C

A-B=\phi

Option C (

A-B=\phi

) is correct because A is a subset of B, meaning all elements of A are also in B, so removing B from A results in an empty set.

29 Easy

29 of 194

Analyzing +4 | -1

Let

P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}

and,

Q=\{\theta: \sin \theta+\cos \theta=\sqrt{2} \sin \theta\}

be two sets. Then,

View answer and explanation

Correct option: D

P=Q

Option D (

P=Q

) is the correct answer because the two sets are equivalent, meaning they contain the same elements.

30 Easy

30 of 194

Application +4 | -1

A = \left\{ {1,\;2,\;3,\;4,\;5} \right\},\;B = \left\{ {2,\;4,\;6} \right\},\;C = \left\{ {3,\;4,\;6} \right\}

, then

\left( {A \cup B} \right) \cap C

View answer and explanation

Correct option: A

\left\{ {3,{\text{ }}4,{\text{ }}6} \right\}

The correct answer is A ({3, 4, 6}) because it is the intersection of the union of A and B with C.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 194 single-correct-answer MCQs on SET THEORY.
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